The matching law was first formulated by R. J. Herrnstein (1961) following an experiment with pigeons on concurrent variable interval schedules. If R1 and R2 are the response rates on two schedules that yield obtained (as distinct from programmed) rates of reinforcement Rf1 and Rf2, the strict matching law holds that the relative reinforcement rate R1/(R1+R2) matches, that is, equals, the relative reinforcement rate Rf1/(Rf1+Rf2). That is,

R1/(R1+R2) = Rf1/(Rf1+Rf2)

This relationship can also be stated in terms of response and reinforcement ratios:

R1/R2 = Rf1/Rf2

Subsequent research has shown that data normally depart from strict matching, but are fitted to a very good approximation by a power function generalisation of the strict matching,

R1/R2 = k(Rf1/Rf2)s

This is more conveniently expressed in logarithmic form

log(R1/R2) = b + s log(Rf1/Rf2)

The constants b and s are referred to as bias and sensitivity respectively. This generalized matching law accounts for high proportions of the variance in most experiments on concurrent variable interval schedules in non-humans. Values of b depend on details of the experiment set up, bu values of s are consistently found to be around 0.8, whereas the value required for strict matching would be 1.0 (Baum, 1974; Davison & McCarthy, 1988).

The matching law is theoretically important for two reasons. First, it offers a simple quantification of behaviour which is capable of extension to a number of other situations. Secondly, it appears to offer a lawful, predictive account of choice; as Herrnstein (1970) expressed it, under an operant analysis, choice is nothing by behavior set into the context of other behavior. It thus challenges any idea of free will, in exactly they way B. F. Skinner had argued that the experimental analysis of behavior should, in his book Beyond freedom and dignity. However this challenge is only serious if the scope of the matching law can be extended from pigeons to humans. When human participants perform under concurrent schedules of reinforcement, matching has been observed in some experiment (e.g. Bradshaw et al, 1976), but wide deviations from matching have been found in others (e.g. Horne & Lowe, 1993).